DYNAMIC PERFORMANCE EVALUATION WITH DEADLINES: THE ROLE OF COMMITMENT

Transcription

1 Discussion Paper No DYNAMIC PERFORMANCE EVALUATION WITH DEADLINES: THE ROLE OF COMMITMENT Chia-Hui Chen Junichiro Ishida October 2017 The Institute of Social and Economic Research Osaka University 6-1 Mihogaoka, Ibaraki, Osaka , Japan

2 Dynamic Performance Evaluation with Deadlines: The Role of Commitment Chia-Hui Chen and Junichiro Ishida October 20, 2017 Abstract We consider an environment in which a principal hires an agent and evaluates his productivity over time in an ongoing relationship. The problem is embedded in a continuoustime model with both hidden action and hidden information, where the principal must induce the agent to exert effort to facilitate her learning process. The value of committing to a deadline is examined in this environment, and factors which make the deadline more profitable are identified. Our framework generates a unique recursive equilibrium structure under no commitment which can be exploited to obtain a full characterization of equilibrium. The analysis allows us to evaluate the exact value of commitment for any given set of parameters and provides insight into when it is beneficial to commit to an evaluation deadline at the outset of a relationship. JEL Classification Number: D82, D83 Keywords: dynamic agency, deadlines, experimentation, commitment, up-or-out contracts. The firstauthor acknowledges financial supportfrom JSPSKAKENHIGrant-in-Aidfor ScientificResearch B) 16H03615 and Young Scientists B) 17K The second author acknowledges financial support from JSPS KAKENHI Grant-in-Aid for Scientific Research S) 15H05728, A) , B) 16H03615 and C) as well as the program of the Joint Usage/Research Center for Behavioral Economics at ISER, Osaka University. Institute of Economic Reserach, Kyoto University. chchen@kier.kyoto-u.ac.jp Institute of Social and Economic Research, Osaka University. jishida@iser.osaka-u.ac.jp

3 1 Introduction Suppose that an employer needs to hire a worker to carry out a project over time. The project is ability-intensive in that the worker can successfully complete the project only if he is sufficiently productive. As is often the case, however, the worker s sheer productivity is not directly observable to the employer, who must instead make an inference from a sequence of observed outputs. Since success cannot be achieved overnight, the employer must exercise some patience even when things do not appear to be in good shape. Excessive tolerance for failure, however, diminishes the worker s incentive to take costly actions that are indispensable to be successful. A number of issues arise in this dynamic environment. How much time should the employer give the worker before she terminates the project? Should the employer commit to a deadline at the outset? If so, under what conditions? In this paper, we attempt to address these issues by analyzing a situation where a principal hires an agent and evaluates his productivity over time in an ongoing relationship. The problem is embedded in a continuous-time framework with both hidden action and hidden information. At each instant, the agent privately chooses how much effort to supply. The outcome is either a success or a failure, depending on his effort choice as well as his productivity type. The game ends immediately when the agent achieves a success or a breakthrough ). The principal s task in this environment is to determine when to terminate the project, conditional on no success having occurred. Within this setup, we analyze two distinct cases to illustrate the role of commitment: in one case, the principal sets a deadline and commits to it at the outset, and the project is terminated automatically when the deadline is reached without attaining a success; in the other, the principal makes no such commitment, thereby retaining discretion to terminate the project at any instant, and simply terminates the project when the continuation payoff is not high enough to justify further experimentation. By directly comparing these two cases, we evaluate the extent to which the principal benefits from committing to an evaluation deadline in this dynamic environment. The driving force of our analysis is a dynamic strategic interaction between the agent s effort choice and the principal s termination strategy. On one hand, the agent s effort choice depends clearly on how much time is left until the project is terminated: since the net value of achieving a success is low when the project is still far from termination, the agent tends to start off with low effort and gradually shift to higher effort as the expected termination date approaches. On the other hand, the principal s willingness to terminate the project depends also on the agent s effort choice: when the agent is less motivated and exerts low effort, less information is revealed about his type, which makes the principal more reluctant 1

4 to terminate the project. This strategic interaction can generate a vicious cycle where the principal s reluctance to terminate the project diminishes the agent s motivation, which in turn makes the principal even more reluctant. It is in general profitable to commit to a deadline when each player s incentive to procrastinate is sufficiently strong. The main contribution of the paper is that we devise an analytical framework that is tractable enough to admit a complete characterization of pure-strategy equilibria, both with and without commitment, while capturing this dynamic interaction. Our framework thus allows us to evaluate the exact value of commitment to an evaluation deadline for any given set of parameters. To this end, our main analytical focus is on the no-commitment case, which is generally far more challenging than the commitment case when dynamic interactions are considered. In the no-commitment case, the principal s termination strategy must be sequentially rational along the way, and the belief off the equilibrium path plays a crucial role. Even in this case, we can show that our framework generates a unique recursive equilibrium structure which can be exploited to establish the existence and uniqueness of pure-strategy) perfect Bayesian equilibrium. We then build on this result to derive a necessary and sufficient condition under which the principal can strictly benefit from committing to an evaluation deadline. As can be expected, the lack of commitment to a deadline entirely alters the dynamic allocation of effort as well as the timing of project termination. When the principal sets a deadlineattheoutset, shedoessobytakingintoaccount howitaffects theagent s entireeffort sequence. In particular, since extending a deadline in general relaxes the agent s incentive compatibility constraint and reduces his total effort supply, the deadline must be set at a point where the expected benefit of achieving a success at the next instant) equals the cost of a decrease in total effort. The situation changes rather drastically when she makes no such commitment, because her termination decision would have no influence on the agent s past behavior. As such, she terminates the project whenever the continuation payoff is about to turn negative while taking the agent s effort strategy as given. This fact implies that the cost of extending a deadline is smaller under no commitment and tends to give her an excessive incentive to wait for a success. Given this incentive structure, one may expect that the principal would always wait longer for a success in equilibrium under no commitment. As it turns out, though, this conjecture does not always hold true in the current setup. Among other things, an interesting, and somewhat counterintuitive, property of our model is that the average duration of the project may not be monotonic with respect to the initial prior belief under no commitment, whereas it is always weakly increasing under commitment. As a direct consequence of this, the average 2

5 duration can be either shorter or longer with commitment than without: in other words, there exists an equilibrium in which the principal prematurely terminates the project when she does not commit to a deadline. This result is somewhat surprising, provided that the inefficiency of the problem stems from the principal s reluctance to terminate the project in the first place. The principal tends to terminate the project too early when the agent s productivity under low effort is sufficiently small. For the sake of argument, suppose that the agent fails almost surely when he exerts low effort, in which case the expected instantaneous payoff is negative while the principal can learn almost nothing from failures during the phase where the agent is supposed to exert low effort). The principal s belief then declines very slowly over time, forcing her to incur a large amount of loss if she is to wait until she totally loses her confidence in the agent s ability. If this expected loss is prohibitively large, the principal may find it optimal to terminate the project even when the belief is still relatively high. Of course, in equilibrium, the agent correctly anticipates this reaction and adjusts his effort allocation accordingly. We show that the equilibrium timing of project termination can be pinned down by backward induction, where we start from the final critical time to be derived) and solve backward. This process gives rise to the aforementioned recursive equilibrium structure which allows us to establish the uniqueness of equilibrium and also obtain a diverse set of equilibrium dynamics under no commitment. The implications of our analysis can be applied broadly to a range of circumstances in which a principal an evaluator) must assess an agent s upside potential that is only gradually revealed in an ongoing relationship, e.g., a manager who must evaluate subordinates, a professor who must evaluate graduate students, a head coach in professional sports who must evaluate players, and so on. Among those possibilities, the most prominent example of evaluation schemes with deadlines is perhaps the up-or-out system, which is widely observed in academia and professional service industries such as law, accounting, and consulting. As a specific application, our framework offers some insight for when up-or-out contracts are more valuable by identifying several key factors such as high ability intensity, stable job descriptions, and similar jobs across ranks which favor the use of an evaluation deadline from a previously unexplored channel. Each of these factors intensifies either the agent s incentive to delay exerting high effort for a given deadline or the principal s incentive to delay terminating the project for a given effort sequence or both), thereby rendering it more profitable to set a deadline at the outset. The paper is organized as follows. The literature review is provided in the remainder of this section. The model is presented in section 2 and analyzed in sections 3 and 4, where we 3

6 characterize both the commitment and no-commitment solutions. These solutions are then compared in section 5, in order to analyze the value of commitment and derive a necessary and sufficient condition under which it is strictly optimal to commit to a deadline. Some extensions of the baseline model and concluding remarks are offered in section 6. Related Literature: The current analysis is most closely related to the experimentation literature in that the principal here attempts to uncover the agent s type through a sequence of experiments. 1 From the principal s point of view, our model can be seen as a variant of the canonical two-armed bandit problem with one safe arm terminating the project) and one risky arm continuing the project). Our model is particularly related to the literature on strategic experimentation which analyzes a situation where a group of individuals, rather than a single individual, face bandit problems Bolton and Harris, 1999; Bergemann and Välimäki, 1996, 2000; Keller et al., 2005; Klein and Rady, 2011; Bonatti and Hörner, 2011). A crucial difference from this strand of literature is that experiments in our context are intermediated, i.e., experiments are conducted not by the principal herself but by an informed intermediary, the agent, who faces no uncertainty about the project. 2 Recently, there have been increasingly many works that explore the optimal provision of incentives in bandit problems. Manso 2011) considers the classic two-armed bandit problem and shows that the optimal contract in this context must tolerate, or even reward, early failures in order to encourage exploratory activities. Bergemann and Hege 1997, 2005) and Hörner and Samuelson 2013) analyze a financing problem of a venture capitalist where the principal provides funding to the agent who conducts experiments on a project of unknown quality. 3 Gerardi and Maestri 2012) consider a similar environment where an agent conducts experiments but assume that the outcome of each experiment can only be observed by the agent. The principal must hence devise a contract not only to induce costly effort but also to truthfully reveal the information. Halac et al. 2013) analyze a model of long-term contracting for experimentation with hidden information about the agent s ability and dynamic moral hazard and obtain an explicit characterization of optimal contracts. Aside from 1 An early economic application of the bandit problem can be found in Rothschild 1974). See Bergemann and Välimäki 2008) for a survey. 2 Several recent works analyze models of delegated experimentation where a principal delegates experimentation to an agent. Guo 2014) analyzes a situation where the principal can specify, with full commitment power, how the agent should allocate the resource in all future contingencies and solves for the optimal delegation rule. Garfagnini 2011) considers a similar setting to ours but assumes that: i) the principal and the agent are symmetrically informed about the state of nature; and ii) the agent s payoff is independent of the state. Chen and Ishida 2015) consider the opposite case in which the principal, with the termination right, may be privately informed about the project quality while the agent focuses on implementing the project assigned to him. 3 Also, see Buisseret 2016) who considers a two-period model of this setting with a more general, convex cost function. 4

7 some technical differences, 4 these previous works are primarily concerned with characterizing optimal contracts. In contrast, the aim of this paper is to compare the allocations under full commitment and no commitment, while considering a less complete contractual environment, in order to evaluate the extent to which the principal can gain from committing to a deadline at the outset. Although it is hard to tell a priori which specification is more plausible, as it depends on the details of the underlying situation, our simple and tractable framework allows us to identify and illuminate the counterintuitive role of commitment in dynamic performance evaluation. Several recent works examine the role of commitment in dynamic moral hazard setting. Mason and Välimäki 2015) consider a dynamic moral hazard problem and derive optimal wage contracts, both with and without commitment on wage payments. Hörner and Samuelson 2016) consider a repeated-game setting in which the principal chooses the scale of the project in addition to contingent payments and characterize the set of equilibrium payoff vectors that can be achieved without commitment. 5 A key difference from our model is that these two works do not consider the principal s learning, which is our main focus, with no hidden information about the agent s type; as a consequence, the project is never terminated in their models. 6 This stands in sharp contrast to our setting in which the principal accumulates information about the agent over time, and the project must be terminated in finite time. 7 It is well known that players often wait until the deadline to reach an agreement in finitehorizon models. This behavior, which is referred to as the deadline effect, is a topic of utmost concern in many bargaining and war of attrition models Hendricks et al., 1988; Spier, 1992; Fershtman and Seidman, 1993; Hörner and Samuelson, 2011; Chen, 2012; Damiano et al., 2012; Fuchs and Skrzypacz, 2013). Some recent works also explore the role of deadlines in dynamic problems with multiple agents. Bonatti and Hörner 2011) analyze a dynamic 4 As a key technical difference, we consider a case where the agent knows his own productivity or the project quality), so that our model belongs to the class of dynamic signaling with stochastic signals), rather than of experimentation, from the agent s point of view. 5 In their model, outcomes are privately observed by the agent, and the moral-hazard problem regards the truthful disclosure of this private information. 6 Georgiadis et al. 2014) analyze the role of commitment in a dynamic contribution games where the manager has the decision right over the project size. With multiple agent types and dynamic learning, Bonatti and Hörner 2017) analyze a symmetric-information) experimentation model in which wages are determined competitively a la Holmström 1999) and characterize effort and wage dynamics with an exogenous termination date. 7 A subtle technical difference which directly arises from this fact lies in the ways in which to construct an equilibrium. Mason and Välimäki 2016) and Hörner and Samuelson 2017) construct an equilibrium via arguments found in the analysis of infinitely repeated games reversion to the worst continuation equilibrium), and as such, their analyses yield a non-degenerate set of equilibria. In our analysis, an equilibrium is obtained via backward induction which gives rise to a unique equilibrium. 5

8 moral-hazard problem with a team in which multiple agents work on a project of unknown quality and briefly discuss the optimal deadline in this context. Campbell et al. 2013) consider a similar environment where two agents work jointly on a project. They assume that there is only one project type but assume that one s own outcomes are his private information. Each agent can exert effort to produce a breakthrough individually, and a successful agent can reveal that he has been successful. The focus of these works is placed on the interaction between the agents, especially the freerider problem, whereas ours is on the dynamic interaction between the principal s termination decisions and the agent s effort choices. 2 A dynamic model of performance evaluation Environment: We employ a continuous-time model because of its greater tractability. Consider a situation where a principal female) hires an agent male) to complete a project. The game ends either when the agent attains a success a breakthrough ) or when the principal terminates the project. The agent is either good with prior probability p 0 0,1) or mediocre with probability 1 p 0. The ability type is the agent s private information and is not directly observable to the principal who must instead evaluate it from a sequence of observed outcomes. Production: Theagent makesunobservableefforta t {l,h}ateachinstantt. 8 Weinterpret that low effort a t = l) refers to the minimum level of effort that can be induced via input monitoring while high effort a t = h) refers to any part of effort that cannot be directly monitoredbyanymeans. Theinstantaneouscost ofeffortaisdenotedbyd a whered h = d > 0 and d l = 0. A success arrives stochastically, depending on the effort choice as well as the agent s type. If the good type chooses a t = a over time [t,t+dt), he attains a success with probability λ a dt where λ h > λ l > 0. In contrast, the mediocre type can never succeed with any effort level. 9 Define λ := λ h λ l. Payoff: We consider an incomplete-contracting environment where contingent rewards on the arrival of a success cannot be enforced. 10 A success yields a net present value of y > 0 8 Our focus on binary effort reflects our implicit presumption that the effort cost and success probability are linear, as usually assumed in this literature e.g., Keller et al, 2005; Bonatti and Horner, 2011, for most of their analysis). As long as this structure is maintained, an extension to continuous effort, say a t [l,h], yields exactly the same allocation and is hence irrelevant. 9 Ourmodelspecification isthusthe breakthrough typeinwhichonesuccess canresolve alltheuncertainty regarding the agent s type an assumption that is predominant in the experimentation literature. See, for instance, Keller et al. 2005) and Bonatti and Hörner 2011, 2017). 10 In reality, we rarely observe complete wage contracts in industries comprised of professionals. For instance, few academic institutions, if any, offer rewards specifically contingent on verifiable measures of output: salaries 6

9 to the principal and b > 0 to the agent; 11 otherwise, they both receive zero. Aside from this, the principal must also pay a flow wage w > 0 to the agent as long as the project continues. 12 We assume that w is exogenously given for most part; the case with an endogenous wage is briefly discussed in the concluding section. The reservation payoff is assumed to be zero for both players. The common discount rate is denoted by r 0, ). Contract: The only contractible decision for the principal in this environment is whether to set a deadline, and if so, at what point. If the principal commits to a deadline, she terminates the project at the deadline but never before) if the agent has not attained a success up to that point. If the principal chooses not to commit to any specific deadline, on the other hand, she retains discretion to terminate the project at any instant. As mentioned earlier, our analytical focus is on the latter case which requires that both players strategies be sequentially rational. 3 Analysis 3.1 Agent s effort decision The agent decides whether or not to exert high effort at each instant. To analyze this problem, it is important to note that the principal s belief affects the agent s payoff only through her termination decision. The agent s optimal effort choice thus depends only on the remaining time to the termination date hereafter, simply the remaining time), i.e., the maximum length of time for which the principal continues the project without attaining a success. The remaining time is obvious when the principal sets a deadline at time τ, in which case the remaining time at time t is simply given by τ t. Even without such an explicit commitment, however, the remaining time can be computed from the principal s equilibrium strategy in essentially the same manner as we shall discuss below. For now, we proceed with the presumption that the remaining time exists and is well-defined at any given point in time. are determined through bilateral negotiations, often dictated by market forces, in some countries whereas they are subject to bureaucratic regulations in others. One possible reason for the lack of complete wage contracts in those industries is that it is often difficult, and perhaps prohibitively costly, to measure the exact value of a success in a verifiable manner. Additionally, this assumption makes our analytical framework applicable to a wider range of circumstances. One such possibility is that the benefit of achieving a success accrues from non-transferrable psychological) gains such as prestige, authority, and the sense of achievement, and extrinsic rewards are hence of secondary importance. There are also many cases where contingent monetary transfers are neither feasible nor desirable, as in a professor-student relationship. 11 Anobviousinterpretationisthaty andbrepresentthecontinuationpayoffsofachievingasuccess, including not only the intrinsic value of a successful outcome but also other benefits of identifying/signaling talent. 12 Alternatively, w can be regarded as a flow cost of production e.g., hiring an agent). Of course, under this interpretation, w is no longer a transfer payment to the agent and does not appear in his payoff. It is straightforward to make the setup consistent with this interpretation without having any qualitative impact on our results. 7

10 The agent s problem is rather straightforward since the mediocre type, knowing that his marginal value of effort is zero, never exerts high effort. As such, we can focus on the good type whom we refer to simply as the agent in what follows. Denote by Uk) the agent s value function when the remaining time is k. Taking k > 0 as given, the value function can be written as Uk) = max λa b d a +w)dt+e rdt 1 λ a dt)uk dt) ). a {l,h} Taking the limit dt 0, we obtain the Bellman equation: ruk) = max λa b d a +w λ a Uk) Uk) ), 1) a {l,h} with lim k 0 Uk) = 0. It is clear from this that the agent chooses high effort if and only if λ b Uk) ) d. As usual in this type of setup see, e.g., Bergemann and Hege, 2005), the continuation payoff Uk) captures the agent s reservation payoff which he receives in case of a failure. The cost of not succeeding today is obviously small when he has a high reservation payoff. Since the continuation payoff is higher when the agent has more time to prove himself, the agent has a stronger incentive to work hard as the project approaches the termination date. This is a manifestation of the deadline effect that lies at the core of our entire analysis. Proposition 1 If d > rb w λ λ l +r, 2) there exists k such that the good type exerts high effort if the remaining time is less than or equal to k and low effort otherwise. The threshold k is given by k = 1 λ h +r ln 1 λ h+r)b d )) λ λ h b d+w if b > d λ, d 0 if λ b. If 2) does not hold, the agent always exerts high effort. Proof: See Appendix. 3) When λ b d, the static incentive is too weak for the agent to exert high effort for any remaining time. In contrast, when 2) fails to hold, the static incentive is strong enough to overcome the dynamic agency cost, and the agent is willing to supply high effort under any circumstance. As these cases only result in trivial solutions and are clearly of less interest for the question we pose here, we restrict our attention to the case where the strength of the static incentive lies in some intermediate range by making the following assumption. 8

11 Assumption 1 λ hb d+w λ h +r > b d λ > 0. Among other things, the assumption implies that the optimal threshold k is bounded from above and away from zero, i.e., k 0, ). 3.2 Principal s termination decision Let p t denote the principal s belief that the agent is good at time t, conditional on no success having occurred. Given some effort sequence {a s } t s=0, the updated belief is then given by p t = p 0 e t 0 λas ds Alternatively, taking the time derivative, we obtain 1 p 0 +p 0 e t 0 λas ds. 4) ṗ t = λ at p t 1 p t ). 5) It is clear that the belief is strictly decreasing over time for any effort choice due to the fact that λ l > 0) until the agent attains a success, in which case the belief immediately jumps up to one. The principal s problem is to determine when to terminate the project, conditional on no success having occurred. Now suppose that the principal intends to terminate the project at time τ > t. Taking the effort sequence as given, the principal s continuation payoff can also be written as a function of the remaining time k and the current belief p t where subject to 5). Vk,p t ) = t+k t λ as p s y w)e s t λau p u+r)du ds, It is immediate to see that for a given effort level, there exists a threshold belief below whichtheprincipal sinstantaneouspayoffisstrictlynegative. Wedenotebyq a := min{ w λ ay,1} the break-even belief at which the instantaneous payoff equals zero under effort a. Combined with the fact that the belief is strictly decreasing over time for any effort sequence, q h represents the absolute lower bound of the belief, as the principal clearly has no incentive to continue the project once her belief dips below this level. Since the belief must reach this level sooner or later due to the fact that λ l > 0 and w > 0), the presence of such a lower bound suggests that the game must end in some finite time. This allows us to solve the game by backward induction. Finally, if the value of a success is too small, the model only admits a trivial solution where the principal chooses to stop immediately or not to hire the agent in the first place). In what follows, therefore, we assume that the value of a success is large enough for the principal to hire the agent at least for some positive duration. 9

12 Assumption 2 p 0 > q h = w λ h y. Since the principal s continuation payoff depends on her belief, it is often convenient for the subsequent analysis to explicitly relate the current and termination beliefs to the remaining time. Suppose that the current belief is p and the principal intends to terminate the project when the belief reaches q. If the agent adopts the best response, the remaining time can be written as Kp,q) such that which reduces to pe λ lmax{kp,q) k,0} λ h min{kp,q),k } 1 p+pe λ lmax{kp,q) k,0} λ h min{kp,q),k } = q, pe λ lkp,q) λ k 1 p+pe λ lkp,q) λ k = q, if Kp,q) > k. In addition, it is also convenient to define a backward operator φq) such that φq)e λ hk 1 φq)+φq)e λ hk = q. The backward operator suggests that if p t = φq) and the agent exerts high effort for t [t,t+k ], then p t+k = q; alternatively, if the agent expects the project to be terminated at p t = q, φq) indicates the belief at which he must switch to high effort. Note that these two notions are closely related in that Kp,q) > k if and only if p > φq). 4 Equilibrium 4.1 The cooperative solution: a benchmark Before we move on to analyze our model, we first examine as a benchmark the case where both the principal and the agent attempt to maximize the sum of their individual payoffs to derive the first-best allocation. More precisely, the instantaneous payoff in the cooperative case, common to both players, is the difference between the expected benefit from a success minus the effort cost, i.e., λ a py +b) d a for a given effort a. 13 The agent s problem is essentially the same and only needs a slight modification. Letting Ũk) denote the agent s continuation payoff, the agent exerts high effort if and only if λ y +b) Ũk) ) d. 13 If the agent shared the same objective as we assume here, his private information could in principle be induced at no cost: given that the agent s reservation payoff is zero, the mediocre type is indifferent between participating and dropping out immediately before time 0), as he would obtain zero payoff in either case for any given deadline. As it turns out, however, the optimal deadline is independent of the initial belief p 0 and the analysis holds irrespective of whether or not the principal can immediately screen out the mediocre type. 10

13 As in the noncooperative case, there exists a threshold k such that the good type exerts high effort if and only if the remaining time is less than or equal to k. Under the maintained assumptions, one can easily verify that the agent is generally more motivated in the cooperative case than in the noncooperative case, i.e., k > k. Given this, it is fairly straightforward to derive the cooperative solution. Since there is no social cost of hiring the mediocre type, the joint surplus from hiring the agent is strictly positive for any belief level, and as such, it is efficient for the principal to continue the project indefinitely until a success is attained. This implies that the good type will eventually succeed at some point, although he only exerts low effort all the way if k is finite. Proposition 2 In the cooperative case, the principal never terminates the project, and the game continues indefinitely. The agent always exerts low effort if and high effort otherwise. d > rb+y) λ λ l +r, 6) Proof: See Appendix. Note the difference between 2) and 6). In the cooperative case, the joint expected payoff of exerting low effort indefinitely is rb+y) λ l +r because the value of a success is now b+y, instead of just b, and there is no flow cost w. 4.2 The commitment solution The principal s problem under commitment is to choose a termination date τ which maximizes herexpectedpayoffattime0whilerestrictingattention todeterministicdeadlines. 14 Wefocus exclusively on deterministic deadlines primarily because it is generally difficult to commit to a lottery, even when it is possible to commit to a particular termination date. To see this point, consideracontract wheretheprincipalterminates theprojectatτ 1 orτ 2, τ 1 < τ 2, each with probability strictly less than one. At time τ 1, however, the principal s preferences are generically strict, i.e., it is strictly better either to stop the continuation payoff is negative) or to continue positive), so that there is no incentive to randomize at that point. If it is strictly better to stop at time τ 1, for instance, the principal stops at time τ 1 with probability one but 14 If stochastic deadlines are feasible, the principal may screen out the low type by offering a menu of contracts, one with a deterministic deadline and the other with stochastic ones. We do not pursue this possibility both because we do not think that it is realistic given the enforcement problem) and because it is outside the scope of our analysis given that our focus is on comparing the commitment and no-commitment solutions). Note that this screening cannot be done if deadlines are deterministic, because both types always strictly prefer a longer deadline. 11

14 knowing that, the contract is effectively reduced to the one with a deterministic deadline. The lack of credible enforcement is perhaps the reason why we almost never observe a contract with stochastic deadlines. As we have already seen, the agent exerts high effort if and only if the remaining time is less than or equal to k. We denote by V k,p t ) the continuation payoff when the agent adopts this best response, where k is the remaining time and p t is the principal s belief. This can be written as V k,p t ) = t+ν t λ l p s y w)e s 0 λ lp u+r)du ds +e t+ν λ t l p u+r)du t+k t+ν λ h p s y w)e s ν λ hp u+r)du ds = p tπ l λ l +r 1 e λl+r)ν )+ p te λl+r)ν π h 1 e λh+r)k ν) ) 1 p t)w 1 e rk ), λ h +r r 7) where ν := max{k k,0} and π a := λ a y w. The first two terms represent the expected gain from the good type whereas the last term represents the expected loss from the mediocre type. Since the termination date is equivalent to the remaining time at time 0, the commitment solution, denoted by τ C, is given by τ C p 0 ) := argmax τ V τ,p 0 ). The continuation payoff is continuous but kinked at τ = k. For τ 0,k ), the first-order condition is given by For τ k, ), it is given by Mτ,p 0 ) := p 0 e λ l+r)τ k ) p 0 π h e λ hτ 1 p 0 )w = 0. 8) ) π l λ l +r λ h +r π h1 e λ h+r)k ) To better understand this condition, it is instructive to rewrite this as 1 p 0 )we rτ = 0. 9) p0 π h e λ lτ k )+λ h k ) 1 p 0 )w ) e rτ = p 0 λ e λ l+r)τ k ) w +ry+πh e λ ) h+r)k. λ h +r Note that the left-hand side is the expected instantaneous payoff for the principal at the time of termination, whereas the right-hand side is the marginal cost of extending the deadline. Since the right-hand side is strictly positive, the principal must set the deadline at a point where the expected instantaneous payoff is positive if τ > k. 12

15 Since the instantaneous payoff is negative for any effort level once the belief dips below q h, there is clearly no incentive to set τ > Kp 0,q h ). When the initial belief is sufficiently close to q h φq h ) p 0 ), it is optimal to let the agent work until the belief reaches this lower bound. When it is far from it p 0 > φq h )), on the other hand, the principal must terminate the project at a point where the belief is still above the lower bound. There are two cases, depending on the value of Mk,p 0 ). 15 If λ l +r λ h +r π h1 e λ h+r)k ) > π l, 10) Mk,p 0 ) < 0 for any p 0 q h,1), and we let ˆp = 1 in this case. If 10) fails to hold, on the other hand, there exists a unique ˆp such that Mk, ˆp) = 0. Then, for p 0 > ˆp, there exists a unique interior solution ˆτp 0 ) such that Mˆτp 0 ),p 0 ) = 0. When the initial prior is in this range, it is not so costly to have a phase where the agent exerts low effort, and as such, the principal would wait beyond time k. Of course, extending the deadline beyond time k means that the agent would slack off at the beginning, thereby pushing back the realization of the high-effort phase. The principal trades off the gain of inducing low effort for an additional instant against the cost of realizing the high-effort phase an instant later and sets the termination date at a point where they are equalized. For clarity, we summarize the set of parameters by Θ := b,y,w,d,λ h,λ l,r). We can then make the following statement. Proposition 3 For any given set of parameters Θ,p 0 ) satisfying Assumptions 1 and 2, there exists a unique commitment solution τ C p 0 ) 0,Kp 0,q h )]. For φq h ) p 0, the commitment solution is given by τ C p 0 ) = Kp 0,q h ). For p 0 > φq h ), there exists some ˆp q l,1] such that τ C p 0 ) = { k if ˆp p 0 > φq h ), ˆτp 0 ) if p 0 > ˆp, where: i) ˆp > q l if 1 > q l ; and ii) ˆp = 1 if and only if 10) holds. The commitment solution is weakly increasing in p 0. Proof: See Appendix. Note that Kp 0,q h ) constitutes the upper bound of τ C p 0 ). Since Kp 0,q h ) is bounded from above when q h > 0, the proposition suggests that the project must be terminated in finite time, meaning that some good projects are bound to be terminated prematurely. This draws clear contrast with the cooperative solution where the project is never terminated. 15 Note that Mk,p 0) < 0 for any k > k and p 0 if and only if Mk,p 0) < 0. 13

16 4.3 The no-commitment solution The situation becomes more complicated, and perhaps more intriguing, when the principal makes no commitment at the outset and hence her termination decision must be sequentially rational. Because of this, the principal s termination strategy now needs to be specified in a different way, as her strategy off the equilibrium path after the project is supposed to be terminated) may matter. More precisely, the principal s strategy is given by a set of termination dates T which consists of a first termination date and further dates along off-equilibrium continuation paths. Formally, for this no-commitment case, we solve for a pure-strategy perfect Bayesian equilibrium hereafter, simply an equilibrium) in which the principal terminates the project with probability one at each termination date a formal definition is given below). 16 To illustrate how we specify the principal s strategy, consider a simple example where the principal s strategy consists of two distinct termination dates, i.e., T = {τ 1,τ 2 } where τ 1 < τ 2. Under this strategy, the principal terminates the project with probability one at each τ i, i = 1,2, conditional on the continuation of the game. 17 This means that the project is surely terminated at time τ 1 on the equilibrium path. To show that this constitutes an equilibrium, however, one must assure that the principal cannot profitably deviate from this strategy. To see this, suppose that the principal unexpectedly continues the project at time τ 1. The game then enters into a new phase where the agent now expects the project to be terminated at the next termination date τ 2 and chooses his effort accordingly a t = h if k τ 2 t and a t = l otherwise). Note also that the principal s belief about the agent s type is not affected by the deviation: given T and the agent s best response, we can still apply Bayes rule to compute how the belief evolves both on and off the equilibrium path. 18 This gives the continuation payoff when the principal deviates which must be non-positive. How long does this process continue? It generally goes on until the principal has no 16 We restrict our attention to pure-strategy equilibria by assuming that the principal terminates the project with probability one when the continuation payoff is zero. Since the principal is actually indifferent at this point, however, there may exist other mixed-strategy equilibria in which the principal terminates the project with probability less than one at each termination date. We do not pursue this possibility for two reasons. First, we do not believe that such an equilibrium is particularly realistic and hence appealing. Second, even with randomization, the game still must end with probability one by time τ 1, and the continuation equilibrium is exactly the same once the game enters the final interval τ 1,τ 2 ); solving backwards, one can see that the structure of equilibrium is essentially the same where the agent starts exerting high effort as t approaches each termination date and switches back to low effort if the project is not terminated at the termination date. 17 If the game continues beyond time τ 2, the principal terminates the project immediately as we will see below. 18 Technically, we assume that the belief depends only on the agent s past effort and is given by 4) even off the equilibrium path. This is actually the no signaling what you don t know condition Fudenberg and Tirole, 1991) although it is a rather obvious restriction in games with two players it only implies that the principal s belief should not be affected by her own deviation). 14

17 incentive to continue the project regardless of the agent s effort choice or, in other words, until the belief reaches the lower bound q h. With slight abuse of notation, we denote by p t the equilibrium belief at time t, conditional on the continuation of the game including the off-equilibrium path). Let τ n denote the time at which p t reaches q h for some n which is unknown at this point). We can then restrict our attention to t 0,τ n ], for it is a dominant strategy for the principal to terminate the project once the belief reaches this lower bound. 19 Lemma 1 In any equilibrium, T is a finite set. Proof: The lemma is directly implied by the fact that any two adjacent termination dates must satisfy τ i τ i 1 k > 0. To see this, suppose otherwise, i.e., k > τ i τ i 1. Suppose further that the principal deviates and continues the project at time τ i 1. Then, it is strictly better for the agent to exert high effort because k > τ i τ i 1. Note, however, that since p t > q h for any t 0,τ n ) by definition, the instantaneous payoff is strictly positive until the belief reaches the next termination belief. This is a contradiction because the principal can strictly benefit from deviating and not stopping at time τ i 1. Given this, it is clear that we can only have finitely many termination dates in 0,τ n ]. Given this result, we denote each element of T by τ i where τ 1 < τ 2 < < τ n for some n. Accordingly, the game is divided into n distinct segments T i, i = 1,2,...,n by termination dates, where T i = τ i 1,τ i ) for i = 1,2,...,n with τ 0 = 0. The remaining time at time t is now given by τ i t for t T i while it is zero for all t > τ n see footnote 19). Let q i := p τ i denote the corresponding termination belief. Only the first segment T 1 is actually played on the equilibrium path, as the game ends with probability one by the time the game reaches time τ 1. For expositional clarity, we call τ 1 the no-commitment solution and denote it by τ NC p 0 ) to indicate its dependence on p 0. The formal definition of our equilibrium is given below. Definition 1 A perfect Bayesian equilibrium in this game is a pair of strategies {a s } s=0 and T := {τ i } n i=1 and a belief system such that: given T, the good-type) agent chooses a t at each t to maximize his continuation payoff, i.e., chooses high effort if and only if the remaining time at time t is less than or equal to k ; 19 Suppose that the principal deviates and continues beyond time τ n. In this case, the principal s instantaneous payoff is negative even if the agent chooses a t = h, so that it is a dominant strategy to terminate immediately for all t > τ n. Given this, since the remaining time is invariably zero, the agent always chooses a t = h. This is the unique continuation equilibrium after time τ n although, to describe this process formally, we need to consider a discrete-time counterpart of our model where there is a minimum time unit, and take the limit as the time unit goes to zero). 15

18 given {a s } s=t and the current belief, the principal terminates the project at time t if and only if the continuation payoff is non-positive; The belief at time t depends only on the effort sequence {a s } t s=0 and is given by 4), both on and off the equilibrium path. To pin down the equilibrium termination dates, we need to pay closer attention to the principal s continuation payoff. Since the agent s best response is the same, the principal s continuation payoff can still bewritten as V k,p t ). Theonly difference is that theprincipal s strategy must now be sequentially rational, which amount to the following two equilibrium conditions which the principal s termination strategy T must satisfy. Condition T: i) q n = q h and ii) if n > 1, for each i = 2,...,n, V Kq i 1,q i ),q i 1 ) = 0. Condition C: For each i = 1,2,...,n, V τ i t,p t ) > 0 for all t T i. Condition T is the usual indifference condition that requires the principal to terminate the project when the continuation payoff when she deviates) is non-positive at each τ i. This indifference condition alone is in general not sufficient because the instantaneous payoff may not be monotonically decreasing over time: in any segment T i, the agent may start off with low effort, during which the instantaneous payoff could be so small that the principal is tempted to stop prematurely. Condition C assures that the principal does not stop before the intended termination date τ i is reached. The next result establishes that there exists a perfect Bayesian equilibrium that is always unique even under no commitment. Proposition 4 For any given set of parameters Θ,p 0 ) satisfying Assumptions 1 and 2, there exists a generically) unique pure-strategy equilibrium. Given Θ, the belief space q h,1) is partitioned into mθ) 1 distinct intervals {P j,p j 1 )} mθ) j=1 where P mθ) = q h and P 0 = 1. For p 0 P j,p j 1 ), j = 1,2,...,mΘ), the equilibrium strategies are as follows: The principal s equilibrium strategy is characterized by a set of n = mθ) j +1 distinct termination dates {τ i } n i=1 where qi = P i+j 1 and τ i τ i 1 > k ; 20 The agent exerts high effort if the remaining time is less than or equal to k and low effort otherwise. Moreover, if mθ) > 1, then q l > P As discussed in footnote 19, a t = 0 for all t τ n. 16

19 Proof: See Appendix A. As the proposition suggests, the structure of equilibrium is thoroughly characterized by how the belief space is partitioned into intervals. A crucial determinant of the partition {P j,p j 1 )} mθ) j=1 is the profitability under low effort which is captured by π l. 21 Below, we first briefly illustrate how we pin down mθ) and {P j,p j 1 )} mθ) j=1 from a given Θ in what follows, we simply denote m = mθ) to save notation). Once they are obtained, it is quite straightforward to derive the equilibrium strategy T for a given p 0. Suppose first that the profitability under low effort is sufficiently high, so that the principal s instantaneous payoff is positive even under low effort. In this case, the principal is not tempted to stop prematurely, and only Condition T is sufficient to pin down the equilibrium. Figure 1 depicts this situation where m = 1: the agent starts off with low effort and switches to high effort when the remaining time is k ; the principal stops when the belief reaches the lower bound P 1 = q h at time τ 1. Formally, m = 1 and hence n = 1 for all p 0 q h,1) if and only if V Kq l,q h ),q l ) > 0, i.e., q l π l λ l +r 1 e λl+r)ν )+ q le λl+r)ν π h 1 e λh+r)k ν) ) > 1 p t)w 1 e rk ), 11) λ h +r r where µ = max{kq l,q h ) k,0} see the proof of Proposition 4, especially Lemma 2, for more detail). Note that 11) depends only on Θ. The no-commitment solution τ NC p 0 ) is monotonic in p 0 if and only if this condition holds. [Figure 1 about here] In contrast, the instantaneous payoff may become negative under low effort when the success probability under low effort is relatively low, in which case we may have a situation where the principal s continuation payoff also becomes negative before the belief reaches q h. Figures 2 and 3 show the evolution of the belief and the expected payoff when m = 2 while fixing the principal s strategy at T = {τ 1 }. In the figures, q l is so high that the instantaneous payoff is negative for the entire interval during which the agent exerts low effort, and there exists a point τ in Figure 3) such that the continuation payoff is negative for t 0,τ ). This implies that T = {τ 1 } does not constitute an equilibrium as it violates Condition C. In this case, the game is divided into two segments, T 1 and T 2, as illustrated in Figure 4. The pair of strategies now satisfies the equilibrium conditions since the instantaneous payoff is always positive in T 1. Formally, if V Kq l,q h ),q l ) < 0, there must exist p q h,q l ) such 21 Note that the partition {P j,p j 1 )} mθ) j=1 is determined solely by Θ while q i depends also on the initial prior p 0. 17

20 that V Kp,q h ),p ) = 0, in which case we redefine P 1 = p and P 2 = q h. As above, m = 2 if and only if V Kq l,p 1 ),q l ) > 0. [Figures 2-4 about here] We can continue this process until we find P 1 such that V Kq l,p 1 ),q l ) > Once the partition is pinned down, we can then easily derive the equilibrium strategies: for p 0 P j,p j 1 ), the principal s equilibrium strategy consists of n = m i+1 termination dates where q i = P i+j 1 and i 1 τ NC = τ 1 = Kp 0,q 1 ), τ i = τ i +Kq i 1,q i ) for i = 2,3,...,n. i =1 Note also that q i 1 > φq i ) > q i because the continuation payoff is always strictly positive when the belief is in q i,φq i )), the range where the remaining time is less than k and the agent exerts high effort. This alternatively means τ i τ i 1 > k, i.e., the length between any two adjacent termination dates must be larger than k. Finally, we would also like to note that the constructed equilibrium is generically unique for any given set of parameters. This uniqueness result stems crucially from the fact that there is a lower bound of the belief q h below which the principal would never continue the project. As stated above, together with the fact that λ l > 0, this implies that the game must end in some finite time, which allows us to solve the game via backward induction analogously to the gap case of the durable-good monopoly problem Fudenberg et al., 1985). More precisely, since the agent s strategy depends only on the remaining time, we know exactly how the game must end as the belief approaches the lower bound. Applying this reasoning backward, we can identify a unique continuation equilibrium for each p t > q h and all the way back to the initial prior p 0. 5 Discussion 5.1 The value of commitment Our framework yields unique commitment and no-commitment solutions which enable us to directly assess the value of commitment to an evaluation deadline. Given that both τ C and τ NC can be written as functions of p 0, the expected equilibrium payoffs can also be written as functions of p 0. Let V C p 0 ) := V τ C p 0 ),p 0 ), V NC p 0 ) := V τ NC p 0 ),p 0 ). 22 When q l = 1, there is no such q 1 and m will go to infinity. If q l < 1, on the other hand, this process must converge after a finite number of rounds. 18